Analysis and geometry on complex homogeneous domains/
Jacques Faraut
- 2000
- New York: Birkhäuser, 2000-01-01.
- 540
I Function Spaces on Complex Semi-groups by Jacques Faraut.- I Hilbert Spaces of Holomorphic Functions.- I.1 Reproducing kernels.- I.2 Invariant Hilbert spaces of holomorphic functions..- II Invariant Cones and Complex Semi-groups.- II.1 Complex semi-groups.- 1I.2 Invariant cones in a representation space.- II.3 Invariant cones in a simple Lie algebra.- III Positive Unitary Representations.- III.1 Self-adjoint operators.- III.2 Unitary representations.- III.3 Positive unitary representations.- IV Hilbert Function Spaces on Complex Semi-groups.- IV.1 Schur orthogonality relations.- IV.2 The Hardy space of a complex semi-group.- IV.3 The Cauchy-Szego kernel and the Poisson kernel.- IV.4 Spectral decomposition of the Hardy space.- V Hilbert Function Spaces on SL(2,?).- V.1 Complex Olshanski semi-group in SL(2,?).- V.2 Irreducible positive unitary representations.- V.3 Characters and formal dimensions of the representations ?m.- V.4 Bi-invariant Hilbert spaces of holomorphic functions.- V.5 The Hardy space.- V.6 The Bergman space.- VI Hilbert Function Spaces on a Complex Semi-simple Lie Group.- VI.1 Bounded symmetric domains.- VI.2 Irreducible positive unitary representations.- VI.3 Characters and formal dimensions.- VI.4 Bi-invariant Hilbert spaces of holomorphic functions.- References.- II Graded Lie Algebras and Pseudo-hermitian Symmetric Spaces by Soji Kaneyuki.- I Semisimple Graded Lie Algebras.- I.1 Root theory of real semisimple Lie algebras.- I.2 Semisimple graded Lie algebras.- I.3 Example.- I.4 Tables.- II Symmetric R-Spaces.- II.1 Symmetric R-spaces and their noncompact duals.- II.2 Sylvester's law of inertia in simple GLA's.- II.3 Generalized conformal structures and causal structures.- III Pseudo-Hermitian Symmetric Spaces.- III.1 Pseudo-Hermitian spaces and nonconvex Siegel domains.- III.2 Simple reducible pseudo-Hermitian symmetric spaces.- References.- III Function Spaces on Bounded Symmetric Domains by Adam Kordnyi.- I Bergman Kernel and Bergman Metric.- I.1 Domains in Cr".- 1.2 Bergman kernel, reproducing kernels.- I.3 The Bergman metric.- II Symmetric Domains and Symmetric Spaces.- II.1 Basic facts, definitions.- II.2 Riemannian symmetric spaces.- II.3 Theory of oiLa's.- II.4 OiLa's of bounded symmetric domains.- II.5 Cartan subalgebras.- III Construction of the Hermitian Symmetric Spaces.- III.1 The Borel imbedding theorem.- III.2 The Harish-Chandra realization.- III.3 Remarks on classification.- IV Structure of Symmetric Domains.- IV.1 Restricted root system, boundary orbits.- IV.2 Decomposition under the Cayley transform.- V The Weighted Bergman Spaces.- V.1 Analysis on symmetric domains.- V.2 Decomposition under K.- V.3 Spaces of holomorphic functions.- VI Differential Operators.- VI.1 Properties of ?s.- VI.2 Invariant differential operators on ?.- VI.3 Further results on $$ \mathbb$$(?).- VI.4 Extending D? to p+.- VII Function Spaces.- VII.1 The holomorphic discrete series.- VII.2 Analytic continuation of the holomorphic discrete series.- VII.3 Explicit formulas for the inner products.- VII.4 L9-spaces and Bergman type projections.- VII.5 Some questions of duality.- VII.6 Further results.- References.- IV The Heat Kernels of Non Compact Symmetric Spaces by Qi-keng Lu.- I Introduction.- II The Laplace-Beltrami Operator in Various Coordinates.- III The Integral Transformations.- IV The Heat Kernel of the Hyperball R?(m, n).- V The Harmonic Forms on the Complex Grassmann Manifold.- VI The Horo-hypercircle Coordinate of a Complex Hyperball.- VII The Heat Kernel of RII(m).- VIII The Matrix Representation of NIRGSS.- References.- V Jordan Triple Systems by Guy Roos.- I Polynomial Identities.- I.1 Definition of Jordan triple systems.- I.2 Identities of minimal degree.- 1.3 Jordan representations and duality.- 1.4 The fundamental identity of degree 7.- 1.5 The Bergman operator.- II Jordan Algebras.- II.1 Jordan algebras arising from a JTS.- II.2 Identities in a Jordan algebra.- II.3 The JTS associated to a Jordan algebra.- III The Quasi-inverse.- III.1 Quasi-invertibility in a Jordan algebra.- 111.2 Quasi-invertibility in a JTS.- 11I.3 Identities for the quasi-inverse.- 1II.4 Differential equations.- 1I1.5 Addition formulas.- IV The Generic Minimal Polynomial.- IV.1 Unital Jordan algebras.- IV.2 General Jordan algebras.- IV.3 Jordan triple systems.- V Tripotents and Peirce Decomposition.- V.1 Tripotent elements.- V.2 Peirce decomposition.- V.3 Orthogonality of tripotents.- V.4 Simultaneous Peirce decomposition.- VI Hermitian Positive JTS.- VI.1 Positivity.- VI.2 Spectral decomposition.- VI.3 Automorphisms.- VI.4 The spectral norm.- VI.5 Classification of Hermitian positive JTS.- VII Further Results and Open Problems.- VII.1 Schmid decomposition.- VII.2 Compactification of an hermitian positive JTS.- VII.3 Projective imbedding.- VII.4 Volume computations.- VII.5 Some open problems.- References.